= Bombieri–Vinogradov theorem
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For every $A>0$ there is $B>0$ such that, for $Q\leq x^{1/2}/\log^Bx$,
$$
\sum_{q\leq Q}\max_{(a,q)=1}\max_{2\leq y\leq x}\left|\psi(y;q,a)-\frac y{\phi(q)}\right|\ll_A\frac{x}{\log^Ax}.
$$
Here $\psi(y;q,a)$ is the <Chebyshev function in an arithmetic progression> and $\phi$ is the <Euler totient function>. <Partial summation> gives the corresponding result for the <prime-counting function> and the offset <logarithmic integral function>. The theorem controls the total error over many moduli; it need not give the same bound for each individual modulus.
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